Step |
Hyp |
Ref |
Expression |
1 |
|
cnvco |
|- `' ( F o. G ) = ( `' G o. `' F ) |
2 |
1
|
imaeq1i |
|- ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( ( `' G o. `' F ) " ( _V \ { Z } ) ) |
3 |
|
imaco |
|- ( ( `' G o. `' F ) " ( _V \ { Z } ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) ) |
4 |
2 3
|
eqtri |
|- ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) ) |
5 |
4
|
imaeq2i |
|- ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) = ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) ) |
6 |
|
funforn |
|- ( Fun G <-> G : dom G -onto-> ran G ) |
7 |
6
|
biimpi |
|- ( Fun G -> G : dom G -onto-> ran G ) |
8 |
7
|
ad2antrl |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> G : dom G -onto-> ran G ) |
9 |
|
simpl |
|- ( ( F e. V /\ G e. W ) -> F e. V ) |
10 |
9
|
anim2i |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( Z e. _V /\ F e. V ) ) |
11 |
10
|
ancomd |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( F e. V /\ Z e. _V ) ) |
12 |
|
suppimacnv |
|- ( ( F e. V /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) ) |
13 |
11 12
|
syl |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) ) |
14 |
13
|
sseq1d |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F supp Z ) C_ ran G <-> ( `' F " ( _V \ { Z } ) ) C_ ran G ) ) |
15 |
14
|
biimpd |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F supp Z ) C_ ran G -> ( `' F " ( _V \ { Z } ) ) C_ ran G ) ) |
16 |
15
|
adantld |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( `' F " ( _V \ { Z } ) ) C_ ran G ) ) |
17 |
16
|
imp |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( `' F " ( _V \ { Z } ) ) C_ ran G ) |
18 |
|
foimacnv |
|- ( ( G : dom G -onto-> ran G /\ ( `' F " ( _V \ { Z } ) ) C_ ran G ) -> ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) ) = ( `' F " ( _V \ { Z } ) ) ) |
19 |
8 17 18
|
syl2anc |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) ) = ( `' F " ( _V \ { Z } ) ) ) |
20 |
5 19
|
syl5eq |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) ) |
21 |
|
coexg |
|- ( ( F e. V /\ G e. W ) -> ( F o. G ) e. _V ) |
22 |
21
|
anim2i |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( Z e. _V /\ ( F o. G ) e. _V ) ) |
23 |
22
|
ancomd |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F o. G ) e. _V /\ Z e. _V ) ) |
24 |
|
suppimacnv |
|- ( ( ( F o. G ) e. _V /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) ) |
25 |
23 24
|
syl |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) ) |
26 |
25
|
imaeq2d |
|- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) ) |
27 |
26
|
adantr |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) ) |
28 |
13
|
adantr |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) ) |
29 |
20 27 28
|
3eqtr4d |
|- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) |
30 |
29
|
exp31 |
|- ( Z e. _V -> ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) ) |
31 |
|
ima0 |
|- ( G " (/) ) = (/) |
32 |
|
id |
|- ( -. Z e. _V -> -. Z e. _V ) |
33 |
32
|
intnand |
|- ( -. Z e. _V -> -. ( ( F o. G ) e. _V /\ Z e. _V ) ) |
34 |
|
supp0prc |
|- ( -. ( ( F o. G ) e. _V /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = (/) ) |
35 |
33 34
|
syl |
|- ( -. Z e. _V -> ( ( F o. G ) supp Z ) = (/) ) |
36 |
35
|
imaeq2d |
|- ( -. Z e. _V -> ( G " ( ( F o. G ) supp Z ) ) = ( G " (/) ) ) |
37 |
32
|
intnand |
|- ( -. Z e. _V -> -. ( F e. _V /\ Z e. _V ) ) |
38 |
|
supp0prc |
|- ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) ) |
39 |
37 38
|
syl |
|- ( -. Z e. _V -> ( F supp Z ) = (/) ) |
40 |
31 36 39
|
3eqtr4a |
|- ( -. Z e. _V -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) |
41 |
40
|
2a1d |
|- ( -. Z e. _V -> ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) ) |
42 |
30 41
|
pm2.61i |
|- ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) |